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7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 223Table 7.5: Levin—Lin Test of PPPNumer- Time Half- Halfaireeffect τ c life τ ct life τc ∗ τct∗yes -8.593 2.953 -9.927 1.796 -1.878 -0.920(0.021) (0.070) (0.164) (0.093)[0.009] [0.074] [0.117] [0.095]US no -6.954 5.328 -7.415 3.943 – –(0.115) (0.651)[0.168] [0.658]yes -8.017 3.764 -9.701 1.816 -1.642 -0.628(0.018) (0.106) (0.154) (0.421)Ger- [0.022] [0.127] [0.158] [0.442]many no -10.252 3.449 -11.185 1.859 – –(0.000) (0.007)[0.001] [0.006]Notes: Bold face indicates signiÞcance at the 10 percent level. Half-lives are basedon bias-adjusted ˆρ by Nickell’s formula [eq.(2.82)] and are stated in years. Nonparametricbootstrap p-values in parentheses. Parametric bootstrap p-values in squarebrackets.because the available time-series are so short. We will revisit the questionby combining observations across the 19 countries that we examinedin the univariate tests into a panel data set. We thus have N =18real exchange rate observations over T = 100 quarterly periods.The results from the popular Levin—Lin test (chapter 2.5) are presentedin Table 7.5. 9 Nonparametric bootstrap p-values in parenthesesand parametric bootstrap p-values in square brackets. τ ct indicates alinear trend is included in the test equations. τ c indicates that only aconstant is included in the test equations. τc∗ and τct ∗ are the adjustedstudentized coefficients (see chapter 2.5). When we account for thecommon time effect, the unit root is rejected at the 10 percent levelboth when a time trend is and is not included in the test equationswhen the dollar is the numeraire currency. Using the deutschemark asthe numeraire currency, the unit root cannot be rejected when a trend9 Frankel and Rose [59], MacDonald [97], Wu [135], and Papell conduct Levin—Lintests on the real exchange rate.